They wouldn't collide at anywhere near C unless they were black holes.

They would collide at their escape velocity, multiplied (I think) by sqrt(2) on account of they're each accelerating the other.

(Earth's escape velocity is about 1m/s at 0.1 light-year. Meaning the entire universe beyond that gives it less than that amount of energy if something's falling from farther away.)

If they are black holes, they would each collide at c relative to the other, and they would each be moving at c relative to an observer at rest at the center of mass, because "2c" doesn't make physical sense as a velocity in any case.

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gmalivuk wrote:They would collide at their escape velocity, multiplied (I think) by sqrt(2) on account of they're each accelerating the other.

Why would escape velocity act as a limit on their final speed? Is it just how the maths works out? A sort-of galactic Achilles and the Tortoise IYSWIM?

At escape velocity, a receding object is, by definition, just fast enough that gravitational attraction will just slow it down to zero velocity at infinity.That implies that an object starting out at zero velocity at "infinity" will just reach escape velocity on impact.

It's less "just how the maths works out" and more "exactly why the math works out that way".

Escape velocity corresponds to a kinetic energy exactly equal to the difference in potential energy between an object's current location and escape (to infinity). Thus it is also the kinetic energy something will have that falls "from infinity" to that point.

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Pretty sure that's how they knew Oumuamua came from outside the solar system. When it zipped by it had more speed than anything that could be bound to the Sun. Or did they actually get enough data to establish the trajectory was hyperbolic?

I mean, if they had enough data to figure out its speed was higher than escape velocity at that distance, that *is* enough to determine its trajectory is hyperbolic.

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I realize either one implies the other. What I don't know is if they directly measured its speed, or got enough data points to approximate its trajectory. But now that I think about it, that would probably also be the same thing. You get one you get the other, right?

Yeah, you could "directly" measure it's speed with the Doppler effect if it was coming straight towards us, I suppose, but otherwise we generally know how fast things are going by noting where they are at different times.

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I know they scanned Oumuamua for radio emissions, but haven't read about any Doppler speed measurements. And you'd need the velocity vector relative to Earth to interpret any Doppler shift.

OK, I'm learning some good shit here. So when they get the 3-4 (?) data points of its location on the celestial sphere, and extrapolate the trajectory from there, w/o knowing the distance, how do they get the speed?

I suspect observations at different times of the day and from different places on Earth allowed for some parallax measurements of distance.

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gmalivuk wrote:They would collide at their escape velocity, multiplied (I think) by sqrt(2) on account of they're each accelerating the other.

Why would escape velocity act as a limit on their final speed? Is it just how the maths works out? A sort-of galactic Achilles and the Tortoise IYSWIM?

At escape velocity, a receding object is, by definition, just fast enough that gravitational attraction will just slow it down to zero velocity at infinity.That implies that an object starting out at zero velocity at "infinity" will just reach escape velocity on impact.

Ah! Thank you - that makes perfect sense.

How can I think my way out of the problem when the problem is the way I think?

Sure, if someone at the center of mass naively adds the apparent velocities together they'll calculate something close to 2c.

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gmalivuk wrote:They would collide at their escape velocity, multiplied (I think) by sqrt(2) on account of they're each accelerating the other.

Why would escape velocity act as a limit on their final speed? Is it just how the maths works out? A sort-of galactic Achilles and the Tortoise IYSWIM?

At escape velocity, a receding object is, by definition, just fast enough that gravitational attraction will just slow it down to zero velocity at infinity.That implies that an object starting out at zero velocity at "infinity" will just reach escape velocity on impact.

Can I just check - by "infinity", do you mean "any distance that's really, really big," or do you mean infinity specifically? If the latter, what if my balls aren't an infinite distance apart? I mean, any return trajectory hits zero at some point.

How can I think my way out of the problem when the problem is the way I think?

But it gets fairly close long before lim x→∞, in Space-Is-Really-Big terms, and a quick back-of-an-imaginary-envelope approximation with the starting conditions of "several thousand light years apart" it probably hardly matters for anything involving "very massive" masses. Whatever order of magnitude that should actually be.

gmalivuk wrote:They wouldn't collide at anywhere near C unless they were black holes.

You could get close to c if they were neutron stars. And of course, if the combined mass & kinetic energy were high enough (after emitting some shrapnel & gravitational waves), the resulting object would very quickly collapse into a black hole.

gmalivuk wrote:They would collide at their escape velocity, multiplied (I think) by sqrt(2) on account of they're each accelerating the other.

I was messing around with this a month ago. I also guessed a factor of sqrt(2), but I was wrong. In the version I was working on, I had a pair of twin Earths falling towards each other.

The terminal speed for a test particle hitting the Earth (with no air resistance) is about 11.18 km/s. That is, when the particle's distance from the centre of the Earth equals the Earth's radius. But with the twin Earths, collision occurs when the distance between their centres equals twice the Earth's radius, and when you factor that in with the fact that they're both accelerating towards each other, you end up with each planet having a speed of 5.59 km/s in the centre of mass frame, so you end up with the same relative collision speed of 11.18 km/s.

"Suppose two identical, uniformly dense, rigid balls of mass M and radius R are initially at rest in a vacuum separated by a distance r. If they interact only gravitationally and relativistic effects are negligible, at what speed v will they collide?"

I think the answer is v = √(GM(r - 2R)/Rr), by simple integration. Of course if r ≫ R, this simplifies to v = √(GM/R), which is the escape velocity over √2. But that's the speed of each body relative to the initial state of rest; the speed of either relative to the other is twice that, since their velocities are equal and opposite, so that relative speed is vrel = 2 √(GM/R) = √2 vescape, as gmal said.

The equation is ΔT = - ΔU (under the assumption of conservation of energy and that only kinetic and gravitational energy are changing, as per the hypothesis). The left side is simply Mv² (where v is relative to the initial rest frame), the sum of the kinetic energies of the moving balls. The right side is -∫2Rr GM²/x² dx. Solving for v should give the answer I have above, unless I made a big mistake.

Soupspoon wrote:But it gets fairly close long before lim x→∞, in Space-Is-Really-Big terms, and a quick back-of-an-imaginary-envelope approximation with the starting conditions of "several thousand light years apart" it probably hardly matters for anything involving "very massive" masses. Whatever order of magnitude that should actually be.

Yeah, as I said above, Earth's escape velocity is 1m/s at about a tenth of a lightyear. Meaning anything falling (from rest) from a greater distance will have at most 1m/s of velocity by the time it reaches that point.

The difference in final velocity on impact, between starting at rest and starting at 1m/s from that distance, is about 45 microns per second. And since said final velocity is 11.2 kilometers per second, I'd say it's pretty safe to ignore that difference.

In other words, because the limit at infinity is easier to calculate than the precise value at some absurd distance, and because the difference it makes is parts per billion for a planet (and even less for something bigger like a star), and because the original question was about velocity and not time, it's safe to treat distances like a thousand light years as infinite, and just use the escape velocity in your calculations.

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For what it's worth, if the spheres are the mass and radius of Earth, the final relative velocity will be 15.8 km/s. If the Sun, 874 km/s. If solar mass white dwarf stars, 8,710 km/s. As gmal said, you only really run into relativity for more compact objects, though I think it first becomes significant for spheres with the mass and radius of neutron stars in this scenario, even before you worry about black holes.